Round edge where multiple faces exists
I come from Lightwave3D and modo, so not that experienced with Blender, but I have tried everything and I just keep getting the same results as you. It looks to me like a limitation of the built-in edge-bevel algorithm, and probably solving with some editing is inevitable.
This is how I would approach this in Blender, given my simple experience so far. I have created a simple demonstration case. Pictures would probably explain the approach better:

I would knife-cut the main face at least, although other connected faces might need the same too (as in my case)

I would then dissolve the faces that are attached to the target edge

and lastly apply the bevel and clean afterwards, if necessary

cheers and good luck,